So, what exactly does all of this mean for Sears, and the company's other big brand, Kmart?

Analysts and pundits have been weighing in on the situation today and they seem worried. Sears was in bad shape to begin with, and this transition presents additional concerns.

Lampert isn't a retail guy.

"While we acknowledge Mr. Lampert's insight and expertise when it comes to 'financial engineering,' we are concerned about his lack of merchandising experience at time when the retailer is struggling to turn around its core Sears and Kmart chains," Gimme Credit senior high yield analyst Evan Mann told Reuters.

"Does Sears now have even more problems attracting top talent in key departments for as long as Lampert remains at the helm?" asked Brian Sozzi, chief equities analyst at NBG Productions. "I am unsure if a creative retail mind would feel confident in jumping ship for fear of being boxed in from an initiative perspective."

But, after all, Lampert has been heavily involved in the inner workings of Sears since he became chairman. So, with D'Ambrosio gone, perhaps nothing will change.

There are concerns that he simply doesn't have any new ideas.

"Maybe he’ll install new carpet at headquarters. Or change the logo," wrote YCharts editor Jeff Bailey. "But it seems doubtful at this late date that the man who has controlled Sears for nearly a decade – and presided over this stock chart – has any new tricks up his sleeve."

"At the end of the day, there is only one person who makes the big decisions ... and that person is Mr. Lampert," wrote Gary Balter, an analyst at Credit Suisse, in a note. "Giving him an additional title does not change that reality, and in our opinion, does not change the direction of the company."

Lampert personally owns 22 percent of Sears Holdings, and his fund, ESL, controls around 40 percent of the company.